Math 121 Test Questions Spring 2010 Chapters 13 and 14 1. (10 pts) The first-semester enrollment at HSC was 1120 students. If all entering freshmen classes were the same size and there were no attrition, then one would expect the 1120 students to be divided equally among freshmen, sophomores, juniors, and seniors. The actual enrollments, by class, are Class Freshman Sophomore Junior Senior Enrollment Actual 314 305 261 240 Expected (a) Fill in the expected enrollments under the above hypothesis. (b) Test the hypothesis that enrollment is uniform across the classes. α = 0.05. Use 2. (10 pts) Normally when one rolls a die, he shakes it up in his hand and then rolls it. That procedure produces a uniform distribution of numbers 1 through 6. I was wondering whether the distribution would still be uniform if I rolled the die without shaking it up first. So I rolled an ordinary die 120 times without shaking it up. Each time, I began with the number 1 facing up and just rolled the die onto the table. The following table shows the results. Number 1 2 3 4 5 6 Count 10 14 16 25 23 32 Test the hypothesis that the results come from a uniform distribution, i.e., the hypothesis that p 1 = p 2 = p 3 = p 4 = p 5 = p 6 = 1, at the 1% level of significance. 6 3. (10 pts) I modified the randint function on my TI-83 in such a way as to simulate a loaded die that lands 1, 2, or 3 each with probability 10%, 4 or 5 each with probability 20%, and 6 with probability 30%. Then I tested my creation by simulating 100 rolls of the loaded die. The results are shown in the following table. Value 1 2 3 4 5 6 Frequency 8 11 14 29 15 23 Test the hypothesis that the distribution from which these numbers were taken is 10%, 10%, 10%, 20%, 20%, 30% for the values 1, 2, 3, 4, 5, 6, respectively. Test at the 1% level of significance. 4. (12 pts) The Richmond Times-Dispatch reported the number of incidents of crime for 24 cities and counties in the central Virginia area for 2007 and 2008. The following table shows the data for a random sample of 10 of those cities and counties. 1
No.of Incidents City/County 2007 2008 Ashland 312 283 Colonial Heights 799 852 Dinwiddie County 485 553 Goochland County 223 234 Hanover County 1140 1189 Henrico County 10518 10795 Hopewell 1166 1302 King William Co. 58 50 Petersburg 2947 3154 Richmond 10971 9759 (a) (10 pts) Use the count data to test the hypothesis that the two distributions are the same from 2007 to 2008, that is, the populations are homogeneous. Test at the 5% level of significance. (b) (2 pts) Under the assumption of homogeneity, find the expected number of incidents of crime in Ashland in 2007. 5. (12 pts) According to an article in the Richmond Times-Dispatch, Nov. 14, 2008, there were 6190 vehicle crashes involving deer in Virginia last year. 1 The article divided the crashes into three categories depending on the type of road: interstate highway, primary road, or secondary road. One might expect the crashes on the interstate to be the most severe and the ones on the secondary roads to be the least severe because of the difference in speed. One might be wrong. The following table shows the number of accidents and the number of injuries in each category. Type of road Interstate Primary Secondary No. of crashes 765 3505 1920 No. of injuries 59 281 145 (a) (10 pts) Test the hypothesis that the two distributions are the same, that is, the populations are homogeneous. Test at the 5% level of significance. (b) (2 pts) Find the expected number of injuries for each category under the assumption of homogeneity. 6. (12 pts) Every year the National Survey of Student Engagement (NSSE) is conducted on all participating campuses. HSC has participated in 2003 and 2007. One of the questions on the 2003 survey was Discussed ideas from your readings or classes with others outside of class and the student responds with Never, Sometimes, Often, or Very Often. The table below shows the results for HSC and for all liberal arts colleges. 1 http://www.inrich.com/cva/ric/search.apx.-content-articles-rtd-2008-11-14-0161.html 2
Never Sometimes Often Very Often HSC 7 54 52 39 All LA colleges 266 2305 3142 2468 (a) (8 pts) Conduct a test, at the 5% level of significance, to determine whether the two populations (HSC students, all LA college students) are homogeneous. (b) (2 pts) According to the null hypothesis, what is the expected number of HSC students who very often discussed ideas from reading with others outside of class? 7. (12 pts) In a sample of 42398 traffic deaths in 2002, investigators determined whether speeding was a factor and whether the road was straight or curved at the site of the accident. They reported the following data. Straight Curved Speeding 8359 5304 Not Speeding 23139 5596 (a) (2 pts) Identify the two variables in this study. (b) (8 pts) Conduct a test, at the 5% level of significance, to determine whether the two variables are independent. (c) (2 pts) According to the null hypothesis, what is the expected number of traffic deaths on straight roads in which speeding was a factor? 8. (25 pts) Is a college s average SAT score a good predictor of what it charges for tuition? A random sample of 10 colleges produced the following data 2. College SAT-Math (x) Tuition (in $1000s) (y) Bennington College 560 33 Calvin College 605 19 Queens College 535 11 Flagler College 560 9 Grove City College 636 10 Indiana Univ. of Pa. 534 12 Lafayette College 665 30 Manhattanville College 550 25 Marlboro College 590 27 Mercer University 587 23 (a) (5 pts) Use these data to find the least-squares regression line. Write the equation. 2 Princeton Review and CNNMoney.com 3
(b) (3 pts) Find the correlation coefficient. (c) (3 pts) What does this correlation coefficient in this problem tell us about the association between these two variables? (d) (4 pts) According to the model, an increase of 10 in the average SAT- Math score for a college would correspond to what change in the expected tuition? (e) (4 pts) The average SAT-Math score at Hampden-Sydney College is 573. Use this value and the model to predict the tuition at HSC. (According to CNNMoney.com, HSC s tuition is $21,878.) (f) (3 pts) In this model, which is the explanatory variable and which is the response variable? (g) (3 pts) Someone might object to this model by pointing out that it does not take into account important factors such as whether the college is private or the college s location (private colleges and colleges in the northeast are typically more expensive). What is the statistical term for these factors? 9. (32 pts) The Project on Student Debt recently published a report on student debt after graduation from college in 2007. The report included figures on the average amount of debt by state (average for Virginia is $18,084) and percentage of students who graduate with debt (percentage for Virginia is 59%). A random sample of 10 states provides the following data. 3 Average Debt Percentage State ($1000s) (x) with Debt (y) Alaska 25.0 53 Alabama 20.9 61 Delaware 17.4 50 Indiana 21.3 60 Kansas 18.5 61 Massachusetts 21.1 63 Ohio 22.0 67 Pennsylvania 23.6 71 Rhode Island 23.2 67 South Carolina 20.2 59 (a) (4 pts) On the graph paper below, draw a scatter plot of the data. (b) (3 pts) Based on your scatter plot, describe the relationship between average debt and percentage of graduates with debt. (c) (5 pts) Find the equation of the regression line. (d) (4 pts) Find the correlation coefficient. What does it tell us about the relationship? 3 www.projectonstudentdebt.org 4
(e) (3 pts) Find the coefficient of determination. What does it tell us about the relationship? (f) (4 pts) Given that SST = 629.6, find SSR and SSE. (g) (4 pts) The average debt in Alabama is $20,921. Use the regression line to predict the percentage of graduates with debt in Alabama. (The actually percentage is 61%.) (h) (5 pts) At the 5% level of significance, test for the significance of the model. That is, test the hypotheses H 0 : β = 0 and ρ = 0 vs. H 1 : β 0 and ρ 0. y 75 70 65 60 55 50 17 18 19 20 21 22 23 24 25 x 5